Remark 2.12 
By Frobenius theorem (see e.g. [1]}), if \(M\)  is a real manifold and
one has an involutive distribution \(L\subset TM\)  then the collection of all integral
manifoldsAn integral manifold of a distribution \(L\subset TM\)  is an immersed submanifold \(N\subset M\)  such
that \(T_{p}N=L|_{p}\)  for all \(p\in N\) . A generic distribution does not admit integral manifolds.
of \(L\)  forms a foliation of \(M\) .
But if the involutive distribution is complex, i.e. \(L\subset TM\otimes \mathbb {C}\) , then the
integral manifolds are complex submanifolds living in the complexification of \(M\) 
Note that, given a real manifold \(M\) , the complex manifold resulting from an integrable almost-complex structure,
and the complexification of \(M\) , are two different concepts..
Since complexification requires real-analyticity of \(M\)  (which is a restrictive property from
the point of view of relativity), in general we will not assume the existence of a foliation.
The involutivity of the distribution, however, is well-defined, and is what we mostly need in this work.

